Let $F$ be a non-Archimedean local field and $O$ be the ring of integers in $F$. Take $G=GL(2,F)$ and $K=GL(2,O)$. Consider the action of $K$ on $G$ by conjugation. Is it possible to explicitly write down the representatives for the K orbits? Is this known?
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$\begingroup$ Use $KA^+K$ decomposition. The orbit space for the left $K$ action is $A^+K$ and the map $kak' \mapsto kak'k^{-1}$ intertwines the left $K$ action with the conjugation $K$-action. This map is the identity on $A^+K$, so this set is also a set of representatives for the conjugation action. $\endgroup$– Uri BaderCommented Jan 31, 2020 at 10:56
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